Integrand size = 178, antiderivative size = 29 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=\frac {2}{-2 x+\frac {4}{\log \left (\frac {x}{\left (16-e^4+\frac {4 x}{3}\right )^2}\right )}} \]
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=\frac {-1-\frac {2}{-2+x \log \left (\frac {9 x}{\left (3 e^4-4 (12+x)\right )^2}\right )}}{x} \]
Integrate[(-96 + 6*E^4 + 8*x + (-48*x + 3*E^4*x - 4*x^2)*Log[(9*x)/(2304 + 9*E^8 + E^4*(-288 - 24*x) + 384*x + 16*x^2)]^2)/(-192*x + 12*E^4*x - 16*x ^2 + (192*x^2 - 12*E^4*x^2 + 16*x^3)*Log[(9*x)/(2304 + 9*E^8 + E^4*(-288 - 24*x) + 384*x + 16*x^2)] + (-48*x^3 + 3*E^4*x^3 - 4*x^4)*Log[(9*x)/(2304 + 9*E^8 + E^4*(-288 - 24*x) + 384*x + 16*x^2)]^2),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (-4 x^2+3 e^4 x-48 x\right ) \log ^2\left (\frac {9 x}{16 x^2+384 x+e^4 (-24 x-288)+9 e^8+2304}\right )+8 x+6 e^4-96}{-16 x^2+\left (16 x^3-12 e^4 x^2+192 x^2\right ) \log \left (\frac {9 x}{16 x^2+384 x+e^4 (-24 x-288)+9 e^8+2304}\right )+\left (-4 x^4+3 e^4 x^3-48 x^3\right ) \log ^2\left (\frac {9 x}{16 x^2+384 x+e^4 (-24 x-288)+9 e^8+2304}\right )+12 e^4 x-192 x} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (-4 x^2+3 e^4 x-48 x\right ) \log ^2\left (\frac {9 x}{16 x^2+384 x+e^4 (-24 x-288)+9 e^8+2304}\right )+8 x+6 e^4-96}{-16 x^2+\left (16 x^3-12 e^4 x^2+192 x^2\right ) \log \left (\frac {9 x}{16 x^2+384 x+e^4 (-24 x-288)+9 e^8+2304}\right )+\left (-4 x^4+3 e^4 x^3-48 x^3\right ) \log ^2\left (\frac {9 x}{16 x^2+384 x+e^4 (-24 x-288)+9 e^8+2304}\right )+\left (12 e^4-192\right ) x}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {-8 x+x \left (4 x-3 e^4+48\right ) \log ^2\left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )+96 \left (1-\frac {e^4}{16}\right )}{x \left (4 x-3 e^4+48\right ) \left (2-x \log \left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{x^2}+\frac {2 \left (-4 x^2+\left (56-3 e^4\right ) x+6 \left (16-e^4\right )\right )}{x^2 \left (4 x-3 e^4+48\right ) \left (2-x \log \left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )\right )^2}+\frac {4}{x^2 \left (x \log \left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )-2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \frac {1}{x^2 \left (x \log \left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )-2\right )^2}dx+4 \int \frac {1}{x^2 \left (x \log \left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )-2\right )}dx+16 \int \frac {1}{\left (-4 x+3 e^4-48\right ) \left (x \log \left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )-2\right )^2}dx+2 \int \frac {1}{x \left (x \log \left (\frac {9 x}{\left (3 e^4-4 (x+12)\right )^2}\right )-2\right )^2}dx-\frac {1}{x}\) |
Int[(-96 + 6*E^4 + 8*x + (-48*x + 3*E^4*x - 4*x^2)*Log[(9*x)/(2304 + 9*E^8 + E^4*(-288 - 24*x) + 384*x + 16*x^2)]^2)/(-192*x + 12*E^4*x - 16*x^2 + ( 192*x^2 - 12*E^4*x^2 + 16*x^3)*Log[(9*x)/(2304 + 9*E^8 + E^4*(-288 - 24*x) + 384*x + 16*x^2)] + (-48*x^3 + 3*E^4*x^3 - 4*x^4)*Log[(9*x)/(2304 + 9*E^ 8 + E^4*(-288 - 24*x) + 384*x + 16*x^2)]^2),x]
3.2.33.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 2.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59
method | result | size |
risch | \(-\frac {1}{x}-\frac {2}{x \left (\ln \left (\frac {9 x}{9 \,{\mathrm e}^{8}+\left (-24 x -288\right ) {\mathrm e}^{4}+16 x^{2}+384 x +2304}\right ) x -2\right )}\) | \(46\) |
norman | \(-\frac {\ln \left (\frac {9 x}{9 \,{\mathrm e}^{8}+\left (-24 x -288\right ) {\mathrm e}^{4}+16 x^{2}+384 x +2304}\right )}{\ln \left (\frac {9 x}{9 \,{\mathrm e}^{8}+\left (-24 x -288\right ) {\mathrm e}^{4}+16 x^{2}+384 x +2304}\right ) x -2}\) | \(69\) |
parallelrisch | \(-\frac {\ln \left (\frac {9 x}{-24 x \,{\mathrm e}^{4}+16 x^{2}+9 \,{\mathrm e}^{8}-288 \,{\mathrm e}^{4}+384 x +2304}\right )}{\ln \left (\frac {9 x}{-24 x \,{\mathrm e}^{4}+16 x^{2}+9 \,{\mathrm e}^{8}-288 \,{\mathrm e}^{4}+384 x +2304}\right ) x -2}\) | \(71\) |
int(((3*x*exp(4)-4*x^2-48*x)*ln(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+ 384*x+2304))^2+6*exp(4)+8*x-96)/((3*x^3*exp(4)-4*x^4-48*x^3)*ln(9*x/(9*exp (4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+(-12*x^2*exp(4)+16*x^3+192* x^2)*ln(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))+12*x*exp(4) -16*x^2-192*x),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (25) = 50\).
Time = 0.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.14 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=-\frac {\log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, {\left (x + 12\right )} e^{4} + 384 \, x + 9 \, e^{8} + 2304}\right )}{x \log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, {\left (x + 12\right )} e^{4} + 384 \, x + 9 \, e^{8} + 2304}\right ) - 2} \]
integrate(((3*x*exp(4)-4*x^2-48*x)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+ 16*x^2+384*x+2304))^2+6*exp(4)+8*x-96)/((3*x^3*exp(4)-4*x^4-48*x^3)*log(9* x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+(-12*x^2*exp(4)+16* x^3+192*x^2)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))+12 *x*exp(4)-16*x^2-192*x),x, algorithm=\
-log(9*x/(16*x^2 - 24*(x + 12)*e^4 + 384*x + 9*e^8 + 2304))/(x*log(9*x/(16 *x^2 - 24*(x + 12)*e^4 + 384*x + 9*e^8 + 2304)) - 2)
Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 0.18 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=- \frac {2}{x^{2} \log {\left (\frac {9 x}{16 x^{2} + 384 x + \left (- 24 x - 288\right ) e^{4} + 2304 + 9 e^{8}} \right )} - 2 x} - \frac {1}{x} \]
integrate(((3*x*exp(4)-4*x**2-48*x)*ln(9*x/(9*exp(4)**2+(-24*x-288)*exp(4) +16*x**2+384*x+2304))**2+6*exp(4)+8*x-96)/((3*x**3*exp(4)-4*x**4-48*x**3)* ln(9*x/(9*exp(4)**2+(-24*x-288)*exp(4)+16*x**2+384*x+2304))**2+(-12*x**2*e xp(4)+16*x**3+192*x**2)*ln(9*x/(9*exp(4)**2+(-24*x-288)*exp(4)+16*x**2+384 *x+2304))+12*x*exp(4)-16*x**2-192*x),x)
Time = 0.32 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=-\frac {2 \, \log \left (3\right ) - 2 \, \log \left (4 \, x - 3 \, e^{4} + 48\right ) + \log \left (x\right )}{2 \, x \log \left (3\right ) - 2 \, x \log \left (4 \, x - 3 \, e^{4} + 48\right ) + x \log \left (x\right ) - 2} \]
integrate(((3*x*exp(4)-4*x^2-48*x)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+ 16*x^2+384*x+2304))^2+6*exp(4)+8*x-96)/((3*x^3*exp(4)-4*x^4-48*x^3)*log(9* x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+(-12*x^2*exp(4)+16* x^3+192*x^2)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))+12 *x*exp(4)-16*x^2-192*x),x, algorithm=\
-(2*log(3) - 2*log(4*x - 3*e^4 + 48) + log(x))/(2*x*log(3) - 2*x*log(4*x - 3*e^4 + 48) + x*log(x) - 2)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (25) = 50\).
Time = 1.44 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.28 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=-\frac {\log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, x e^{4} + 384 \, x + 9 \, e^{8} - 288 \, e^{4} + 2304}\right )}{x \log \left (\frac {9 \, x}{16 \, x^{2} - 24 \, x e^{4} + 384 \, x + 9 \, e^{8} - 288 \, e^{4} + 2304}\right ) - 2} \]
integrate(((3*x*exp(4)-4*x^2-48*x)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+ 16*x^2+384*x+2304))^2+6*exp(4)+8*x-96)/((3*x^3*exp(4)-4*x^4-48*x^3)*log(9* x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))^2+(-12*x^2*exp(4)+16* x^3+192*x^2)*log(9*x/(9*exp(4)^2+(-24*x-288)*exp(4)+16*x^2+384*x+2304))+12 *x*exp(4)-16*x^2-192*x),x, algorithm=\
-log(9*x/(16*x^2 - 24*x*e^4 + 384*x + 9*e^8 - 288*e^4 + 2304))/(x*log(9*x/ (16*x^2 - 24*x*e^4 + 384*x + 9*e^8 - 288*e^4 + 2304)) - 2)
Time = 13.54 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {-96+6 e^4+8 x+\left (-48 x+3 e^4 x-4 x^2\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )}{-192 x+12 e^4 x-16 x^2+\left (192 x^2-12 e^4 x^2+16 x^3\right ) \log \left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )+\left (-48 x^3+3 e^4 x^3-4 x^4\right ) \log ^2\left (\frac {9 x}{2304+9 e^8+e^4 (-288-24 x)+384 x+16 x^2}\right )} \, dx=-\frac {2}{x\,\left (2\,x\,\ln \left (3\right )-x\,\ln \left (384\,x-288\,{\mathrm {e}}^4+9\,{\mathrm {e}}^8-24\,x\,{\mathrm {e}}^4+16\,x^2+2304\right )+x\,\ln \left (x\right )-2\right )}-\frac {1}{x} \]
int(-(8*x + 6*exp(4) - log((9*x)/(384*x + 9*exp(8) + 16*x^2 - exp(4)*(24*x + 288) + 2304))^2*(48*x - 3*x*exp(4) + 4*x^2) - 96)/(192*x - 12*x*exp(4) - log((9*x)/(384*x + 9*exp(8) + 16*x^2 - exp(4)*(24*x + 288) + 2304))*(192 *x^2 - 12*x^2*exp(4) + 16*x^3) + 16*x^2 + log((9*x)/(384*x + 9*exp(8) + 16 *x^2 - exp(4)*(24*x + 288) + 2304))^2*(48*x^3 - 3*x^3*exp(4) + 4*x^4)),x)